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Theorem sge0tsms 46335
Description: Σ^ applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0tsms.g 𝐺 = (ℝ*𝑠s (0[,]+∞))
sge0tsms.x (𝜑𝑋𝑉)
sge0tsms.f (𝜑𝐹:𝑋⟶(0[,]+∞))
Assertion
Ref Expression
sge0tsms (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))

Proof of Theorem sge0tsms
Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2734 . . . 4 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )
21a1i 11 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
3 xrltso 13179 . . . . . 6 < Or ℝ*
43supex 9500 . . . . 5 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V
54a1i 11 . . . 4 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V)
6 elsng 4644 . . . 4 (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
75, 6syl 17 . . 3 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
82, 7mpbird 257 . 2 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
9 sge0tsms.x . . . . . . 7 (𝜑𝑋𝑉)
109adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
11 sge0tsms.f . . . . . . 7 (𝜑𝐹:𝑋⟶(0[,]+∞))
1211adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13 simpr 484 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 46328 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
1511ffnd 6737 . . . . . . . . 9 (𝜑𝐹 Fn 𝑋)
1615adantr 480 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
17 fvelrnb 6968 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1816, 17syl 17 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1913, 18mpbid 232 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
20 iccssxr 13466 . . . . . . . . . . . . . 14 (0[,]+∞) ⊆ ℝ*
21 sge0tsms.g . . . . . . . . . . . . . . 15 𝐺 = (ℝ*𝑠s (0[,]+∞))
22 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
2311adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
24 elinel1 4210 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
25 elpwi 4611 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2624, 25syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
2726adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
28 fssres 6774 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥𝑋) → (𝐹𝑥):𝑥⟶(0[,]+∞))
2923, 27, 28syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
30 elinel2 4211 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3130adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
32 0red 11261 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
3329, 31, 32fdmfifsupp 9412 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) finSupp 0)
3421, 22, 29, 33gsumge0cl 46326 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ (0[,]+∞))
3520, 34sselid 3992 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
3635ralrimiva 3143 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
37363ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
38 eqid 2734 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))
3938rnmptss 7142 . . . . . . . . . . 11 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
4037, 39syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
41 snelpwi 5453 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
42 snfi 9081 . . . . . . . . . . . . . . 15 {𝑦} ∈ Fin
4342a1i 11 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ Fin)
4441, 43elind 4209 . . . . . . . . . . . . 13 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
45443ad2ant2 1133 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
4611adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → 𝐹:𝑋⟶(0[,]+∞))
47 snssi 4812 . . . . . . . . . . . . . . . . . . 19 (𝑦𝑋 → {𝑦} ⊆ 𝑋)
4847adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → {𝑦} ⊆ 𝑋)
4946, 48fssresd 6775 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
5049feqmptd 6976 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
51 fvres 6925 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹𝑥))
5251mpteq2ia 5250 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))
5352a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5450, 53eqtrd 2774 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5554oveq2d 7446 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
56553adant3 1131 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
57 xrge0cmn 21443 . . . . . . . . . . . . . . . . 17 (ℝ*𝑠s (0[,]+∞)) ∈ CMnd
5821, 57eqeltri 2834 . . . . . . . . . . . . . . . 16 𝐺 ∈ CMnd
59 cmnmnd 19829 . . . . . . . . . . . . . . . 16 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
6058, 59ax-mp 5 . . . . . . . . . . . . . . 15 𝐺 ∈ Mnd
6160a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐺 ∈ Mnd)
62 simp2 1136 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
6311ffvelcdmda 7103 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ (0[,]+∞))
64633adant3 1131 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ (0[,]+∞))
65 dfss2 3980 . . . . . . . . . . . . . . . . . 18 ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
6620, 65mpbi 230 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
6766eqcomi 2743 . . . . . . . . . . . . . . . 16 (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
68 ovex 7463 . . . . . . . . . . . . . . . . 17 (0[,]+∞) ∈ V
69 xrsbas 21413 . . . . . . . . . . . . . . . . . 18 * = (Base‘ℝ*𝑠)
7021, 69ressbas 17279 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
7168, 70ax-mp 5 . . . . . . . . . . . . . . . 16 ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
7267, 71eqtri 2762 . . . . . . . . . . . . . . 15 (0[,]+∞) = (Base‘𝐺)
73 fveq2 6906 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
7472, 73gsumsn 19986 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑦𝑋 ∧ (𝐹𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
7561, 62, 64, 74syl3anc 1370 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
76 simp3 1137 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
7756, 75, 763eqtrrd 2779 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
78 reseq2 5994 . . . . . . . . . . . . . 14 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
7978oveq2d 7446 . . . . . . . . . . . . 13 (𝑥 = {𝑦} → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
8079rspceeqv 3644 . . . . . . . . . . . 12 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
8145, 77, 80syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
82 pnfxr 11312 . . . . . . . . . . . . 13 +∞ ∈ ℝ*
8382a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ℝ*)
8438elrnmpt 5971 . . . . . . . . . . . 12 (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8583, 84syl 17 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8681, 85mpbird 257 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
87 supxrpnf 13356 . . . . . . . . . 10 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
8840, 86, 87syl2anc 584 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
89883exp 1118 . . . . . . . 8 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9089adantr 480 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9190rexlimdv 3150 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦𝑋 (𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞))
9219, 91mpd 15 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
9314, 92eqtr4d 2777 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
949adantr 480 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
9511adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
96 simpr 484 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
9795, 96fge0iccico 46325 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
9894, 97sge0reval 46327 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
9923, 27feqresmpt 6977 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
10099adantlr 715 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
101100oveq2d 7446 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))))
10221fveq2i 6909 . . . . . . . . . . 11 (+g𝐺) = (+g‘(ℝ*𝑠s (0[,]+∞)))
103 eqid 2734 . . . . . . . . . . . . . 14 (ℝ*𝑠s (0[,]+∞)) = (ℝ*𝑠s (0[,]+∞))
104 xrsadd 21414 . . . . . . . . . . . . . 14 +𝑒 = (+g‘ℝ*𝑠)
105103, 104ressplusg 17335 . . . . . . . . . . . . 13 ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞))))
10668, 105ax-mp 5 . . . . . . . . . . . 12 +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))
107106eqcomi 2743 . . . . . . . . . . 11 (+g‘(ℝ*𝑠s (0[,]+∞))) = +𝑒
108102, 107eqtr2i 2763 . . . . . . . . . 10 +𝑒 = (+g𝐺)
10921oveq1i 7440 . . . . . . . . . . 11 (𝐺s (0[,)+∞)) = ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞))
110 icossicc 13472 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ (0[,]+∞)
11168, 110pm3.2i 470 . . . . . . . . . . . 12 ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
112 ressabs 17294 . . . . . . . . . . . 12 (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞)))
113111, 112ax-mp 5 . . . . . . . . . . 11 ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
114109, 113eqtr2i 2763 . . . . . . . . . 10 (ℝ*𝑠s (0[,)+∞)) = (𝐺s (0[,)+∞))
11558elexi 3500 . . . . . . . . . . 11 𝐺 ∈ V
116115a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
117 simpr 484 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
118110a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
119 0xr 11305 . . . . . . . . . . . . 13 0 ∈ ℝ*
120119a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ∈ ℝ*)
12182a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → +∞ ∈ ℝ*)
12295ad2antrr 726 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
12326sselda 3994 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
124123adantll 714 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
125122, 124ffvelcdmd 7104 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
12620, 125sselid 3992 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ*)
127 iccgelb 13439 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑦))
128120, 121, 125, 127syl3anc 1370 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ≤ (𝐹𝑦))
129 id 22 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑦) = +∞ → (𝐹𝑦) = +∞)
130129eqcomd 2740 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) = +∞ → +∞ = (𝐹𝑦))
131130adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ = (𝐹𝑦))
13211ffund 6740 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → Fun 𝐹)
133132ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → Fun 𝐹)
13422, 123sylan 580 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
13511fdmd 6746 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → dom 𝐹 = 𝑋)
136135eqcomd 2740 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 = dom 𝐹)
137136ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
138134, 137eleqtrd 2840 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
139 fvelrn 7095 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
140133, 138, 139syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
141140adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ ran 𝐹)
142131, 141eqeltrd 2838 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
143142adantl3r 750 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
14496ad3antrrr 730 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
145143, 144pm2.65da 817 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → ¬ (𝐹𝑦) = +∞)
146145neqned 2944 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
147 ge0xrre 45483 . . . . . . . . . . . . . 14 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
148125, 146, 147syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
149148ltpnfd 13160 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) < +∞)
150120, 121, 126, 128, 149elicod 13433 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
151 eqid 2734 . . . . . . . . . . 11 (𝑦𝑥 ↦ (𝐹𝑦)) = (𝑦𝑥 ↦ (𝐹𝑦))
152150, 151fmptd 7133 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)):𝑥⟶(0[,)+∞))
153 0e0icopnf 13494 . . . . . . . . . . 11 0 ∈ (0[,)+∞)
154153a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
155 eliccxr 13471 . . . . . . . . . . . 12 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
156 xaddlid 13280 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
157 xaddrid 13279 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
158156, 157jca 511 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
159155, 158syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
160159adantl 481 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
16172, 108, 114, 116, 117, 118, 152, 154, 160gsumress 18707 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
162 rege0subm 21458 . . . . . . . . . . . . 13 (0[,)+∞) ∈ (SubMnd‘ℂfld)
163162a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
164 eqid 2734 . . . . . . . . . . . 12 (ℂflds (0[,)+∞)) = (ℂflds (0[,)+∞))
165117, 163, 152, 164gsumsubm 18860 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
166 eqidd 2735 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
167 vex 3481 . . . . . . . . . . . . . 14 𝑥 ∈ V
168167mptex 7242 . . . . . . . . . . . . 13 (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V
169168a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V)
170 ovexd 7465 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂflds (0[,)+∞)) ∈ V)
171 ovexd 7465 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠s (0[,)+∞)) ∈ V)
172 rge0ssre 13492 . . . . . . . . . . . . . . . . 17 (0[,)+∞) ⊆ ℝ
173 ax-resscn 11209 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℂ
174172, 173sstri 4004 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℂ
175 cnfldbas 21385 . . . . . . . . . . . . . . . . 17 ℂ = (Base‘ℂfld)
176164, 175ressbas2 17282 . . . . . . . . . . . . . . . 16 ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂflds (0[,)+∞))))
177174, 176ax-mp 5 . . . . . . . . . . . . . . 15 (0[,)+∞) = (Base‘(ℂflds (0[,)+∞)))
178177eqcomi 2743 . . . . . . . . . . . . . 14 (Base‘(ℂflds (0[,)+∞))) = (0[,)+∞)
179110, 20sstri 4004 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ*
180 eqid 2734 . . . . . . . . . . . . . . . 16 (ℝ*𝑠s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
181180, 69ressbas2 17282 . . . . . . . . . . . . . . 15 ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞))))
182179, 181ax-mp 5 . . . . . . . . . . . . . 14 (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞)))
183178, 182eqtri 2762 . . . . . . . . . . . . 13 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞)))
184183a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞))))
185 rge0srg 21473 . . . . . . . . . . . . . . 15 (ℂflds (0[,)+∞)) ∈ SRing
186185a1i 11 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (ℂflds (0[,)+∞)) ∈ SRing)
187 simpl 482 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
188 simpr 484 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
189 eqid 2734 . . . . . . . . . . . . . . 15 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℂflds (0[,)+∞)))
190 eqid 2734 . . . . . . . . . . . . . . 15 (+g‘(ℂflds (0[,)+∞))) = (+g‘(ℂflds (0[,)+∞)))
191189, 190srgacl 20222 . . . . . . . . . . . . . 14 (((ℂflds (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
192186, 187, 188, 191syl3anc 1370 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
193192adantl 481 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
194172a1i 11 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
195 id 22 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
196195, 178eleqtrdi 2848 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
197194, 196sseldd 3995 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ ℝ)
198197adantr 480 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ ℝ)
199172a1i 11 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
200 id 22 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
201200, 178eleqtrdi 2848 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
202199, 201sseldd 3995 . . . . . . . . . . . . . . 15 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ ℝ)
203202adantl 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ ℝ)
204 rexadd 13270 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
205204eqcomd 2740 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
206162elexi 3500 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ∈ V
207 cnfldadd 21387 . . . . . . . . . . . . . . . . . . . . 21 + = (+g‘ℂfld)
208164, 207ressplusg 17335 . . . . . . . . . . . . . . . . . . . 20 ((0[,)+∞) ∈ V → + = (+g‘(ℂflds (0[,)+∞))))
209206, 208ax-mp 5 . . . . . . . . . . . . . . . . . . 19 + = (+g‘(ℂflds (0[,)+∞)))
210209, 207eqtr3i 2764 . . . . . . . . . . . . . . . . . 18 (+g‘(ℂflds (0[,)+∞))) = (+g‘ℂfld)
211210, 207eqtr4i 2765 . . . . . . . . . . . . . . . . 17 (+g‘(ℂflds (0[,)+∞))) = +
212211oveqi 7443 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
213212a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
214180, 104ressplusg 17335 . . . . . . . . . . . . . . . . . . 19 ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞))))
215206, 214ax-mp 5 . . . . . . . . . . . . . . . . . 18 +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞)))
216215eqcomi 2743 . . . . . . . . . . . . . . . . 17 (+g‘(ℝ*𝑠s (0[,)+∞))) = +𝑒
217216oveqi 7443 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
218217a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
219205, 213, 2183eqtr4d 2784 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
220198, 203, 219syl2anc 584 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
221220adantl 481 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
222 funmpt 6605 . . . . . . . . . . . . 13 Fun (𝑦𝑥 ↦ (𝐹𝑦))
223222a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦𝑥 ↦ (𝐹𝑦)))
224150, 177eleqtrdi 2848 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
225224ralrimiva 3143 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
226151rnmptss 7142 . . . . . . . . . . . . 13 (∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
227225, 226syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
228169, 170, 171, 184, 193, 221, 223, 227gsumpropd2 18705 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
229165, 166, 2283eqtrd 2778 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
23030adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
231148recnd 11286 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
232230, 231gsumfsum 21469 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
233229, 232eqtr3d 2776 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
234101, 161, 2333eqtrrd 2779 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (𝐺 Σg (𝐹𝑥)))
235234mpteq2dva 5247 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
236235rneqd 5951 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
237236supeq1d 9483 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
23898, 237eqtrd 2774 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
23993, 238pm2.61dan 813 . . 3 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
24021, 9, 11, 1xrge0tsms 24869 . . 3 (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
241239, 240eleq12d 2832 . 2 (𝜑 → ((Σ^𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )}))
2428, 241mpbird 257 1 (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  wrex 3067  Vcvv 3477  cin 3961  wss 3962  𝒫 cpw 4604  {csn 4630   class class class wbr 5147  cmpt 5230  dom cdm 5688  ran crn 5689  cres 5690  Fun wfun 6556   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  Fincfn 8983  supcsup 9477  cc 11150  cr 11151  0cc0 11152   + caddc 11155  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293   +𝑒 cxad 13149  [,)cico 13385  [,]cicc 13386  Σcsu 15718  Basecbs 17244  s cress 17273  +gcplusg 17297   Σg cgsu 17486  *𝑠cxrs 17546  Mndcmnd 18759  SubMndcsubmnd 18807  CMndccmn 19812  SRingcsrg 20203  fldccnfld 21381   tsums ctsu 24149  Σ^csumge0 46317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231  ax-mulf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xadd 13152  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-rest 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-ordt 17547  df-xrs 17548  df-mre 17630  df-mrc 17631  df-acs 17633  df-ps 18623  df-tsr 18624  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18966  df-minusg 18967  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-ur 20199  df-srg 20204  df-ring 20252  df-cring 20253  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-ntr 23043  df-nei 23121  df-cn 23250  df-haus 23338  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-tsms 24150  df-sumge0 46318
This theorem is referenced by: (None)
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